R61 tested to n=5000.

3 (probable) primes found, 3 remain.

(probable) primes:

(10*61^1552-1)/3
(13*61^4134-1)/12
(77*61^3080-1)/4

The remain k's are 37, 53, 100.

Found a (probable) prime for S37: (19*37^5310+1)/4.

This (probable) prime is the largest (probable) prime I found for this project (extended SR problems).

S37 has now only 1 k remain, and this k is a half GFN ((37*37^n+1)/2). Thus, the "weak S37 conjecture" is proven!!!

No (probable) prime found for the 5 k's remain for R31: 5, 19, 51, 73, 97. All these k's are likely tested to n=5000.

Reserve S61.

S61 is proven!!!

3 (probable) primes were found.

(23*61^1659+1)/12
(43*61^2788+1)/4
(62*61^3698+1)/3

A harder base was proven!!!

:bow::crazy::lavalamp:

Reserve S31.

[QUOTE=sweety439;457213]
A harder base was proven!!!
[/QUOTE]
Harder than blowing a bubblegum bubble? Yes. But barely.

S10 k=269 at n=24.8K, R7 k=197 at n=29.3K, both no (probable) prime found.

No (probable) prime found for S31, 11 k's still remain. All the k's are likely tested to n=5000.

Update newer zip file to include recent status for S37, S61 and R61.

Reserve S3 to n=6K.

S3 tested to n=6K.

11 (probable) primes found, 27 remain.

(probable) primes:

(821*3^5512+1)/2
(935*3^3967+1)/2
(2909*3^3058+1)/2
(4495*3^4297+1)/2
(5147*3^5153+1)/2
(6041*3^3544+1)/2
(6967*3^3797+1)/2
(7297*3^3234+1)/2
(8389*3^3510+1)/2
(9061*3^4836+1)/2
(9721*3^5040+1)/2

The remain k's are 621, 823, 1187, 1801, 2747, 3007, 3047, 3061, 3307, 5321, 5743, 5893, 6427, 6569, 6575, 7927, 8161, 8227, 8467, 8609, 8863, 8987, 9263, 9449, 10207, 10243, 10741.

S3, k=621 at n=10K, no (probable) prime found.